Chaitin \Omega numbers and halting problems

TL;DR

This paper explores the properties of Chaitin's 3a number, its relation to the halting problem, and the computational power of its base-two expansion under finite restrictions, extending the understanding of algorithmic randomness.

Contribution

It investigates the relative computational power of 3a numbers and halting problems with finite size constraints, elaborating on their Turing equivalence.

Findings

3a number's initial bits solve halting problems for inputs up to that length

3a number and halting problem are Turing equivalent

Finite restrictions reveal nuanced computational relationships

Abstract

Chaitin [G. J. Chaitin, J. Assoc. Comput. Mach., vol.22, pp.329-340, 1975] introduced \Omega number as a concrete example of random real. The real \Omega is defined as the probability that an optimal computer halts, where the optimal computer is a universal decoding algorithm used to define the notion of program-size complexity. Chaitin showed \Omega to be random by discovering the property that the first n bits of the base-two expansion of \Omega solve the halting problem of the optimal computer for all binary inputs of length at most n. In the present paper we investigate this property from various aspects. We consider the relative computational power between the base-two expansion of \Omega and the halting problem by imposing the restriction to finite size on both the problems. It is known that the base-two expansion of \Omega and the halting problem are Turing equivalent. We thus consider an elaboration of the Turing equivalence in a certain manner.